The rapid proliferation of electric vehicle car is fundamentally reshaping the energy landscape. As millions of EVs integrate into power grids, they present not only a substantial new load but also a vast, distributed energy resource with remarkable flexibility. The inherent fast response and significant aggregated capacity of EV aggregators make them ideal candidates for providing crucial grid services, such as frequency regulation and peak shaving, thereby enhancing grid stability and facilitating the integration of intermittent renewable sources. However, unlocking this potential is hampered by critical challenges stemming from the inherent uncertainties of user behavior and the computational complexity of managing large-scale fleets.
Effectively modeling and controlling an electric vehicle car aggregator requires a nuanced understanding of diverse user requirements. Primarily, each user has a fundamental travel energy demand, characterized by a target state of charge (SOC) needed for their next trip. Secondly, users exhibit varied regulation preferences; some may be willing to allow bidirectional power flow (V2G), while others may only permit unidirectional charging control or decline participation altogether. Lastly, growing concerns over data protection limit the granularity of personal and vehicular data that can be centrally collected. These多元化的需求 introduce significant uncertainties into the aggregator’s available regulation capacity. Furthermore, the conventional approach of modeling thousands of individual electric vehicle car dynamics and summing their time-series power profiles results in a high-dimensional, complex model. This “curse of dimensionality” leads to prohibitive computational delays, rendering real-time control and market participation infeasible for large-scale fleets. This paper addresses these intertwined problems by proposing a novel dimensionality reduction modeling framework combined with a robust control strategy, ensuring both accurate capacity estimation and stable power output during regulation services.
The foundation of our approach is a detailed model of the individual electric vehicle car, accounting for its interaction with the grid and user-defined constraints. An EV can be in one of three states: charging (drawing power), idle (no power exchange), or discharging (feeding power to the grid). The core dynamic is the evolution of its battery State of Charge (SOC). For an electric vehicle car $i$ at time $t$, the SOC $S_i(t)$ is given by:
$$S_i(t + \Delta t) = S_i(t) + \frac{P_i(t) \eta_i(t) \Delta t}{Q_i}$$
subject to $S_i(t^{\text{in}}_i) = S^{\text{in}}_i$.
Here, $P_i(t)$ is the power exchange (positive for charging, negative for discharging), $\eta_i(t)$ is the charging/discharging efficiency, $Q_i$ is the battery capacity, $t^{\text{in}}_i$ is the grid connection time, and $S^{\text{in}}_i$ is the initial SOC. The power exchange depends on the connection state $\tau_i(t)$:
$$P_i(t) =
\begin{cases}
P^c_i & \text{if } \tau_i(t) = 1 \text{ (charging)}\\
0 & \text{if } \tau_i(t) = 0 \text{ (idle)}\\
-P^d_i & \text{if } \tau_i(t) = -1 \text{ (discharging)}
\end{cases}$$
where $P^c_i$ and $P^d_i$ are the rated charging and discharging power, often assumed equal.
The user’s travel energy need is defined by a target SOC $S^{\text{set}}_i$ with a tolerance $\delta_i$, forming an acceptable energy interval $[S^L_i, S^H_i]$ where $S^L_i = S^{\text{set}}_i – \delta_i/2$ and $S^H_i = S^{\text{set}}_i + \delta_i/2$. The control logic must ensure the SOC remains within or returns to this interval. User regulation preference categorizes them into three types: Type 1 (non-participants, charge to $S^{\text{set}}_i$ and idle), Type 2 (participate in charging/idle transitions only), and Type 3 (full participants in V2G, including all state transitions). To protect battery health, the depth of discharge (DOD) is constrained, which effectively limits the width of the user’s acceptable SOC interval: $S^H_i – S^L_i \leq D^{\text{DOD}}_{\text{max}}$, typically set to 0.1 to minimize degradation.
To address data privacy, only aggregated statistical parameters are shared with the central aggregator, not individual user data. The local data $\Phi_{\text{con}}$ and uploaded data $\Phi_{\text{up}}$ are:
$$\begin{aligned}
\Phi_{\text{con}} &= \{ I_i, t^{\text{fsh}}_i, S^{\text{set}}_i, \delta_i, P^c_i, Q_i, \eta_i \} \\
\Phi_{\text{up}} &= \{ \mu_P, \sigma_P, \mu_Q, \sigma_Q, \mu_\eta, \sigma_\eta, \mu_{S^{\text{set}}}, \sigma_{S^{\text{set}}}, \mu_\delta, \sigma_\delta, \gamma_i(t) \}
\end{aligned}$$
where $\mu$ and $\sigma$ denote the mean and standard deviation of the respective parameters across the fleet, and $\gamma_i(t)$ indicates connection status.
The key parameters for an electric vehicle car, including its energy and travel patterns, are subject to uncertainties. Their distributions, derived from real-world data and studies, are summarized below.
| Parameter | Distribution | $\mu$ Distribution | $\sigma$ Distribution |
|---|---|---|---|
| $P^c_i / P^d_i$ (kW) | $N(\mu, \sigma^2)$ | $U(7, 12)$ | $U(0.5, 2)$ |
| $Q_i$ (kWh) | $N(\mu, \sigma^2)$ | $U(45, 60)$ | $U(0.01, 0.1)$ |
| $\eta_i$ | $N(\mu, \sigma^2)$ | $U(0.90, 0.93)$ | $U(0.01, 0.1)$ |
| $S^{\text{set}}_i$ | $N(\mu, \sigma^2)$ | $U(0.61, 0.78)$ | $U(0.07, 0.12)$ |
| $\delta_i$ | $N(\mu, \sigma^2)$ | $U(0.05, 0.08)$ | $U(0.05, 0.1)$ |
| Parameter | Distribution |
|---|---|
| Grid connection time $t^{\text{in}}_i$ | $N(6.5, 3.4^2)$ for $t \in [0, 5.5)$; $N(17.5, 3.4^2)$ for $t \in [5.5, 24)$ |
| Grid departure time $t^{\text{fsh}}_i$ | $N(8.9, 3.4^2)$ for $t \in [0, 20.9)$; $N(32.9, 3.4^2)$ for $t \in [20.9, 24)$ |
| Initial SOC $S^{\text{in}}_i$ | $N(0.42, 0.05^2)$ for $S \in [0.2, 0.4]$ |
To overcome the high-dimensionality challenge, we employ the High Dimensional Model Representation (HDMR) method. HDMR approximates a complex, nonlinear system’s output $f(\mathbf{x})$ as a sum of functions of increasing input variable dimensionality:
$$f(\mathbf{x}) \approx f_0 + \sum_{i=1}^{n} f_i(x_i) + \sum_{1 \leq i < j \leq n} f_{ij}(x_i, x_j)$$
where $\mathbf{x} = [x_1, x_2, …, x_n]^T$ is the normalized input vector (e.g., statistical parameters $\mu_P, \sigma_{S^{\text{set}}}$, etc.), $f_0$ is a constant, $f_i$ represents the first-order effect of $x_i$, and $f_{ij}$ represents the second-order interaction between $x_i$ and $x_j$. Higher-order terms are neglected. The component functions are approximated using orthogonal polynomials, such as Legendre polynomials:
$$f_i(x_i) = \sum_{r=1}^{k} \alpha_r^i \phi_r(x_i), \quad f_{ij}(x_i, x_j) = \sum_{p=1}^{l}\sum_{q=1}^{m} \beta_{pq}^{ij} \phi_p(x_i)\phi_q(x_j)$$
The coefficients $\alpha_r^i$ and $\beta_{pq}^{ij}$ are determined via least-squares regression on a large sample of Monte Carlo simulations. For each sample, fleet parameters are randomly drawn from their distributions, the charging/discharging behavior of thousands of electric vehicle cars is simulated under the proposed control logic, and the aggregate upward $P^{\text{up}}$ and downward $P^{\text{low}}$ regulation capacities are computed as outputs.
HDMR provides not only a meta-model but also global sensitivity indices. The first-order sensitivity index $F_i$ measures the fractional contribution of input $x_i$ to the total output variance:
$$D_i \approx \sum_{r=1}^{k} (\alpha_r^i)^2, \quad F_i = \frac{D_i}{D}$$
where $D$ is the total variance. This analysis reveals which uncertain parameters most significantly impact the electric vehicle car aggregator’s regulation capacity, allowing for a focused and reduced-order model.
Our analysis identifies three dominant parameters: the mean rated power ($\mu_P$), the mean target SOC ($\mu_{S^{\text{set}}}$), and the standard deviation of the initial SOC ($\sigma_S$). The fitted HDMR models for a 10,000-electric vehicle car fleet, using third-order polynomials, are:
$$\begin{aligned}
P^{\text{up}} &\approx -0.690 + 8.220\phi_1(\mu_P) – 4.602\phi_1(\mu_{S^{\text{set}}}) – 0.675\phi_1(\sigma_S) + 3.171\phi_1(\mu_P)\phi_1(\mu_{S^{\text{set}}}) + \ldots \\
P^{\text{low}} &\approx 0.690 – 0.689\phi_1(\mu_P) + 1.066\phi_1(\mu_{S^{\text{set}}}) – 0.722\phi_1(\sigma_S) – 2.076\phi_1(\mu_P)\phi_1(\mu_{S^{\text{set}}}) + \ldots
\end{aligned}$$
where $\phi_1(x)=\sqrt{3}(2x-1)$. The sensitivity indices confirm the dominance of these parameters, with $F_{\mu_P}$ and $F_{\mu_{S^{\text{set}}}}$ each around 0.28 for both capacities, and $F_{\sigma_S}$ around 0.12.
A critical issue during regulation is power output instability. When an electric vehicle car’s SOC hits the boundary of its user-defined interval $[S^L_i, S^H_i]$, it must switch states (e.g., from discharging to idle) to satisfy the user’s travel need. If many electric vehicle cars reach their boundary simultaneously, the aggregate power can exhibit a sudden, large deviation from the regulation signal, causing a secondary disturbance. To prevent this, we propose a control method that dynamically adjusts the effective operational boundaries for subsets of electric vehicle cars.
The core idea is to classify electric vehicle cars based on the direction of the required regulation power change $\Delta P_r(t)$ (increase or decrease) and their current SOC relative to adaptive boundaries. For a regulation demand to decrease power ( $\Delta P_r(t) < 0$ ), we temporarily expand the lower boundary $S^L_i$ to $S^{Ll}_i = S^L_i + \Delta S^L_i$ (where $\Delta S^L_i < 0$) for selected electric vehicle cars with sufficient regulation margin, allowing them to discharge more deeply without immediately triggering a state switch. Conversely, for electric vehicle cars close to their boundary, we restrict their participation. A symmetric logic applies for power increase demands ( $\Delta P_r(t) > 0$ ) by adjusting the upper boundary $S^H_i$ to $S^{Hh}_i = S^H_i + \Delta S^H_i$. The control signals $\Delta S^L_i$ and $\Delta S^H_i$ are calculated by the aggregator to precisely meet the regulation signal $\Delta P_r(t)$ while considering the real-time available capacity $P^{\text{low}}(t)$ or $P^{\text{up}}(t)$ predicted by the HDMR model.
The control sequence is as follows: 1) The aggregator receives $\Delta P_r(t)$. 2) It queries the HDMR model for available capacity $P^{\text{low}}(t)$ or $P^{\text{up}}(t)$. 3) It determines the actual dispatch signal $\Delta P_{\text{er}}(t) = \max(\Delta P_r(t), P^{\text{low}}(t))$ (for downward regulation). 4) Electric vehicle cars are sorted according to their SOC and user preference. 5) The aggregator calculates the necessary boundary adjustment $\Delta S^L_i$ for the minimal set of electric vehicle cars whose combined power change matches $\Delta P_{\text{er}}(t)$:
$$\Delta S^L_i(t) = -\frac{|P^d_i| \Delta t \eta^d_i}{Q_i} + \lambda, \quad \text{for } i \in \Omega(t)$$
where $\Omega(t)$ is the ordered set of selected electric vehicle car indices, and $\lambda$ is a small threshold. This method ensures smooth power tracking by proactively managing state transitions.
We validate the proposed framework through comprehensive case studies simulating a fleet of 10,000 electric vehicle cars. First, the accuracy of the HDMR-based capacity evaluation model is tested. The model is trained on 10,000 samples and tested on 100 unseen scenarios. The results show a high degree of alignment between the HDMR-predicted and Monte Carlo-simulated regulation capacities.
| Number of EVs in Sample | Avg. Relative Error for $P^{\text{up}}$ | Avg. Relative Error for $P^{\text{low}}$ |
|---|---|---|
| 5,000 | 10.11% | 9.09% |
| 10,000 | 4.70% | 3.92% |
| 15,000 | 3.47% | 2.87% |
| 20,000 | 2.64% | 2.16% |
The method is robust across different user preference mixes, maintaining errors below 5%.
| User Preference Mix (Type1:Type2:Type3) | Avg. Error for $P^{\text{up}}$ | Avg. Error for $P^{\text{low}}$ |
|---|---|---|
| 0:0:1 (All V2G) | 4.70% | 3.92% |
| 0:1:0 (Charge/Idle only) | 5.21% | 4.17% |
| 1:3:6 (Mixed) | 4.81% | 3.55% |
Second, the impact of dominant uncertain parameters is isolated. Simulations confirm that varying only $\mu_P$, $\mu_{S^{\text{set}}}$, and $\sigma_S$ produces the full range of capacity variation observed when all parameters vary, validating the HDMR sensitivity results. The regulation capacity scales nearly linearly with $\mu_P$ and has a quadratic relationship with $\mu_{S^{\text{set}}}$.
Finally, the integrated control strategy is tested. We compare three scenarios over a one-hour period where the electric vehicle car aggregator is required to follow a downward regulation signal:
- S1: Baseline (No regulation).
- S2: Regulation using the simulated maximum capacity (ideal benchmark).
- S3: Regulation using the proposed HDMR-based capacity prediction and boundary control method.
The results demonstrate that S3 successfully tracks the regulation signal with stability comparable to the ideal benchmark (S2), effectively eliminating the large power deviations that would occur from uncontrolled boundary hits. The dynamic available capacity $P^{\text{low}}(t)$ is also accurately reflected.
The computational efficiency of the HDMR approach is transformative. Compared to the conventional “Monte Carlo + time-series summation” method, the HDMR model reduces simulation time by over 99.4% for large fleets, as it replaces thousands of differential equations with a single algebraic meta-model.
| Number of EVs | Simulation Time: Monte Carlo | Simulation Time: HDMR |
|---|---|---|
| 5,000 | 933.44 s | 5.58 s |
| 10,000 | 4219.86 s | 11.82 s |
| 15,000 | 8430.38 s | 16.63 s |
| 20,000 | 14695.26 s | 19.97 s |
In conclusion, this work presents a holistic solution for the practical management of large-scale electric vehicle car aggregators. By explicitly modeling user-centric uncertainties—travel needs, regulation preferences, and data privacy—and employing the HDMR technique for drastic dimensionality reduction, we achieve an accurate, computationally efficient assessment of the aggregator’s real-time regulation capacity. Furthermore, the novel boundary control strategy ensures stable power output during service provision, preventing secondary grid disturbances. This integrated modeling and control framework paves the way for reliable, scalable, and real-time participation of the burgeoning electric vehicle car fleet in modern power grid operations, turning a potential challenge into a cornerstone for a flexible and sustainable energy system.